Bargaining in the Shadow of Chapter 7: The Consequences of Separating Default and Bankruptcy (Preliminary Draft & Do not cite)

Transcription

1 Bargaining in the Shadow of Chapter 7: The Consequences of Separating Default and Bankruptcy (Preliminary Draft & Do not cite) David Benjamin SUNY Buffalo and Xavier Mateos-Planas Queen Mary University of London February 5, 2 Abstract This paper studies informal default in consumer credit as the start a process of negotiation with the lender. We consider an economy with uninsurable individual risk where households in default bargain with their creditors while preserving the option of declaring formal bankruptcy. Our first result is that a costly procedure like Chapter 7 occurs in equilibrium, owing to commitment issues inside negotiations. Notably, debtors can use negotiations as a vehicle to run down their, impeding the ability of creditors to eventually collect in Chapter 7. The possibility of bargaining accounts for a large fraction of borrowing and default. Informal defaulters are notably wealthier and hold more debt than formal defaulters. They borrowed heavily in anticipation of likely negotiations afterwards. Thus aggregate debt is higher than in models without bargaining. Including bargaining also has qualitative consequences for welfare analysis. A tighter asset exemption in bankruptcy increases the rate of informal default and improves welfare. Finally, we present welfare results about some much discussed institutions including collection policy outside bankruptcy and attempts to encourage negotiations into debtor-friendly settlements. Keywords: consumer unsecured credit, bankruptcy, bargaining, default x.mateos-planas@qmul.ac.uk.

2 bargaining in the shadow of chapter 7 Introduction This paper investigates the process of informal default in consumer credit. In the U.S., default outside formal bankruptcy procedures is common. This is the theme of recent empirical and theoretical studies in Dawsey and Ausubel (24), Dawsey, Hynes, and Ausubel (28) and Chatterjee (2). In the Survey of Consumer Finances 27, about % of the U.S. population had filed under Chapter 7, whereas over 5% held delinquent loans. This is suggestive that informal bankruptcy might account for the bulk of loan write-offs, a point already made in Dawsey and Ausubel (24). This evidence notwithstanding, only formal bankruptcy filings have attracted analysis and discussion in the recent macroeconomic literature. Given its apparent significance, disregarding informal bankruptcy might be a serious oversight in discussions of policies that relate to bankruptcy. This paper takes the view of informal default as a process of negotiation between the debtor and the creditor that may ensue after debt repayments fail to be met. This includes intense negotiations through lawyers and bank administrators as well as unserious negotiations that are simply attempts to keep collection agents at bay. The objectives of this paper are as follows. We put forward an equilibrium model where bargaining is an alternative to formal Chapter 7 bankruptcy procedures. We then draws some potentially testable implications regarding the characteristics of households taking the alternative default options and the settlement outcomes. We show that the model is consistent with a broad amount of data. Finally we assess the significance of bargaining for the positive and welfare consequences of policy changes. We compare these to the outcomes and welfare analysis generated by more traditional models. We show that the policy implications of our model are qualitatively different from models without negotiations and quantitatively strong. The theoretical framework is based on a version of the equilibrium model of heterogeneous households with uninsurable risk typical of much bankruptcy literature like Chatterjee, Corbae, Nakajima, and Ríos-Rull (27) suitably extended to account for two novel features, bargaining and adjustment costs for. The latter feature allows us to consider households who maintain both debt and, a missed feature of many such models in the literature. 2 We include this feature because the portfolio mix matters for the outcome of negotiations. This is because the present value and face value of debt diverges during the negotiations. The former feature is more novel and the core contribution of the paper. Default can take the form of a formal procedure modeled on the U.S. bankruptcy code including a level of asset exemption or can lead to a process of bargaining. The bargaining, which involves complete information, takes place through randomly alternating offers between the debtor and the creditors. The two bargain over a plan by which the debtor For example, Chatterjee, Corbae, Nakajima, and Ríos-Rull (27), Athreya, Tam, and Young (29), Livshits, MacGee, and Tertilt (27), Mateos-Planas (29) and Mateos-Planas and Ríos-Rull (2). Chatterjee has a non-circulating draft which considers collection but not negotiations outside formal default. The two produce different outcomes. 2 Li and Sarte (26) and Hintermaier and Koeniger (29) are notable exceptions.

3 bargaining in the shadow of chapter 7 2 can avoid Chapter 7 and rejoin the market. A plan consists of proposed transfers of current resources and debt swaps which are priced market rates. It also involves a recommendation for allowable consumer next period. Failed negotiations can continue indefinitely or revert to Chapter 7. Debt prices in equilibrium reflect the risk of default and recovery rates for each type of household in terms of income, savings and borrowing. We solve the model for a numerical benchmark. In this initial draft, we seek a setting suitable for learning the main insights from the model. Some parameters are assigned directly, and we focus on the costs of defaulting to broadly match selected empirical targets from the Survey of Consumer Finances, including the formal and informal default rates, the debt to income ratio, the proportion of population in debt and the average interest rate on unsecured loans. The focus of the analysis is on the properties of the stationary distribution of households over characteristics. We find the following properties for the benchmark economy. Our first result is that Chapter 7 occurs in equilibrium at all. Our model is a model of complete information bargaining. This means that the outcomes of bargaining are Pareto Optimal. Thus one would expect that a costly procedure like Chapter 7 is never used in equilibrium. However Chapter 7 occurs for commitment issues inside default. Creditors are afraid the debtors will use Chapter 7 to run down their, hampering future collection efforts. Thus they collect aggressively and force debtors into Chapter 7. Debtors with low are afraid that creditors will use negotiations to delay their entry into Chapter 7 which extends their exclusion from credit markets. This leads such debtors to voluntarily enter Chapter 7. Our secondary results concern the quantitative implications of the model. The possibility of bargaining accounts for a large fraction of total debt in the economy. Defaulters who bargain are notably wealthier and hold more debt than Chapter 7 filers. They exploit in their favor the opportunity to borrow heavily and then default when their exceed the bankruptcy exemption level. Although higher interest rates price in the risk for lenders, they do not deter this loans given that negotiations will result in partial recovery at a time that is convenient to the debtors. This sort of opportunistic motivation of informal defaulters is in contrast with the motivation for formal defaulters. The latter, holding only low, do fully discharge their debts. Markets price them out of large loans as a consequence. The outcomes from negotiations can unstable as individuals default repeatedly in short periods of time. Many defaulters start out in negotiations, reach temporary settlement while exhausting their, default on their settlements and end up in Chapter 7. There is delay in settlements too as it may be efficient to wait till the household rebuilds its after a period of difficulties. Our model is broadly consistent with the differences in the data between the two groups of defaulters. It is also consistent with the data on the evolution of debt and for individuals entering and exiting default. We conduct some basic policy experiments. We show that ruling out informal bankruptcy,

4 bargaining in the shadow of chapter 7 3 as in much of the literature, has substantial consequences. Indebtedness declines as the incentive for opportunistic borrowing behaviors disappears. The consequence of this is that large unobservable penalties must be included to reproduce debt levels in the data, which affects quantitative welfare results. The formal default rate does not vary much, confirming that there is not much substitutability between the two types of default choices in the short term. In terms of welfare, removing the option to bargain causes a generalized welfare loss, especially for highly indebted households. This indicates that consumers benefit from the richer set of borrowing opportunities that are allowed in a model with renegotiation and that commitment is of limited usefulness. The exemption level of under bankruptcy plays an important role in the economy with bargaining. Individuals whose asset accumulation has them far below the law s exemption tend to default quickly into Chapter 7 repaying none of their debt. Individuals who are closer to the exemption level tend to respond to bad shocks by borrowing heavily and accumulating far in excess of the exemption level. A tighter exemption increases the levels of debt and in the economy. The rate of informal default rises notably as this change expands the range of asset position where households exploit the opportunity to borrow much and then bargain. Tightening the exemption level brings about substantial welfare gains at all debt levels. This is qualitatively different then the corresponding result in a model without renegotiation, where little behavior is affected by the exemption level. There are also many frequently discussed institutions which this model can capture, but for which traditional models are silent about. In particular there has been much recent discussion about building negotiation procedures that are favorable to debtors, changing collection law, or affecting adjustment costs by relaxing or tightening consumers access to credit. 3 This paper is a first attempt to modeling default both outside and inside of formal bankruptcy. It is related to the consumer bankruptcy literature. Chatterjee, Corbae, Nakajima, and Ríos-Rull (27), Athreya, Tam, and Young (29) and Livshits, MacGee, and Tertilt (27) consider formal default only without renegotiation. Mateos-Planas and Ríos-Rull (2) is a model of long-term credit contracts with switching costs which could be interpreted as a form of renegotiation, although the process is not one of explicit bilateral bargaining. Although Li and Sarte (26) add Chapter 3 bankruptcy to Chapter 7, the arrangement is not subject to negotiation. We should say that the favor of our results is different than the ones previously described. In the previously described papers, unless the income process involves very persistent shocks, policy analysis prefers tough default penalties that discourage default. Our results support policies that encourage, at the least, informal default. In the sovereign literature, Eaton and Gersovitz (98) and Arellano (28) discuss the behavior of short term debt and, more recently, Benjamin and Wright (29) have first noted 3 Upcoming versions of the paper will contain a more formal version of this section.

5 bargaining in the shadow of chapter 7 4 that including bargaining can sharply improve the ability of these models to reproduce data outcomes. However the inclusion of Chapter 7 produces a richer set of outcomes than the results in that model. In particular we consider the very real possibility that consumers can use negotiations to run down the value of their, a policy which doesn t make sense to countries. 2 Model Our theory concerns the borrowing, lending, and negotiations between private lenders and their creditors. The formal model has two types of agents. The first is infinitely lived risk-averse households who face idiosyncratic income risk. They can hold debt and to smooth consumption. They choose whether to default or not, whether to default via Chapter 7 or via bargaining, and, if given the chance, which terms of settlement to propose and, otherwise, whether to accept the bank s offered terms. The other type of agents is risk neutral creditors who lend competitively to the debtor. When bargaining banks choose, if given the chance, which terms of settlement to propose and, otherwise, whether to accept the household s offered terms. Individual states. At the beginning of each period, a household is described by an idiosyncratic productivity state s, asset holdings a, debt holdings b, and credit status z. There are four possible situations in terms of credit status for the household z {,, 2, 3}. If z =, the household has a clean record and can freely borrow and lend or default; if z =, the household had filed for or been under Chapter 7 bankruptcy in the previous period and not cleared its bankruptcy flag; if z = 2 the household had entered bargaining in the previous period and not reached a settlement and the household is the current proposer; if z = 3 the household had entered bargaining in the previous period and not reached a settlement and the lender is the current proposer. Preferences. Preferences are defined over sequences of consumption c over the household s lifetime. At any point, it is represented as the expected sum of period utility u(c) discounted at the rate β. The utility function belongs in the CRRA class and is specified as u(c) = c σ σ, where σ is the inverse of the rate of risk aversion. Utility is also affected by a non-pecuniary cost to defaulting on debt via Chapter 7 denoted χ 7, and via bargaining denoted χ B. These costs enter separably and additively in the utility of the household. For the most part we will assume they have a common value χ d. There is ex-post individual random noise in the preferences over some choices in some states in the form of an additive and separable shocks to utility. The purpose is primarily technical

6 bargaining in the shadow of chapter 7 5 as this achieves some needed smoothness, although these shocks lend themselves to intuitive interpretation. When z =, there is a random component to the utility of defaulting which is normally distributed with zero mean and standard deviation σ DvsR. This is picking up unobservable heterogeneity in factors that affect the default decision. When having to decide between Chapter 7 default or bargaining that is, when z = and having decided to default or when z = 2, 3 there is a random component to the utility of opting for Chapter 7 which is normally distributed with zero mean and standard deviation σ 7vsB. Technology. Household income e is given by the idiosyncratic productivity type s. The income type is a Markov process with transition matrix Γ e,e over a set E κ. Changing the level of from a to a incurs an adjustment quadratic cost of φ a (a a ) 2. Banks operate an intermediation technology of deposits into one-period loans. They face a transaction cost at maturity which is a proportion λ of debt outstanding. Markets. The households buy perishable consumption c in competitive markets at a price normalized to. They can hold non-contingent debt b and a. Households and banks interact in two environments, in the market and, when in default, outside of the market. Consider the market here. Households buy and banks sell deposits at the risk-free discount price q. Households sell and banks buy debt b at a discount price that depends on all the relevant information: and debts carried forward, a and b, and the current income state s. Thus the menu of debt prices is written as q(a, b, s). Bankruptcy and bargaining. There is imperfect enforcement of contracts. Failure of debts to be repaid may place households and banks outside of the market, being either in Chapter 7 bankruptcy or in a process of bargaining. There affairs will be conducted under specific institutional conditions. Let us consider the timing first. If the household has a clear record z =, it is when it defaults that the exact identity of the would-be proposer, whether the household or the bank, becomes known. Then there is the choice about how to default, whether by Chapter 7 or bargaining. If Chapter 7 follows, then the Chapter 7 flag carries over into the next period z = with probability π 7, otherwise z =. If bargaining follows, then the household will start clean next period z = if they reach a settlement, or will carry a bargaining flag z = 2 or 3 if there is no settlement, depending on which party will be the next the proposer. If the household has currently a past-bargaining flag z = 2, 3 the identity of the would-be proposer is known by definition, and events unfold just like in the previous case, starting with the choice whether to file for Chapter 7 or continue the bargaining. Finally, if the household has a Chapter 7 flag z =, it is again brought over into the next period z = with probability π 7, otherwise z =. From the above description, the decision whether to repay or default in the first place is made before the bargaining plan s proposer is known, but any later decision on the modality of default is made in that knowledge. The identity of the proposer is stochastic. Without default history, it is the result of an independent draw and γ ho denotes the ex-ante probabil-

7 bargaining in the shadow of chapter 7 6 ity that the household is the party who would propose. Once with a bargaining flag, there may be persistence in the proposer s identity, and γ ho,ho and γ lo,lo denote the probability of repeating proposer in this case. We now turn to the institutional rules governing the different default situations. Consider first the case where the household is under Chapter 7 bankruptcy. The household s above a certain exemption level, ā 7, will be seized by banks as repayment, up to a value equivalent to debt outstanding. Banks get to obtain only a fraction γ 7 of those, a description of the efficiency of the recovery technology. Households cannot borrow but can save. The probability of clearing the bankruptcy flag, π 7, describes the punishment to the household in terms of exclusion from the credit market. Consider next the case where the household is in a process of bargaining. The designated party proposes a plan involving new debt b and current transfer of goods τ and, if the bank proposes, new a. If there is no settlement, the bank uses a exogenous collection process whereby it recovers a fraction given by the collection tax on debt τ b. In this situation, the household is free to manage its. If there is settlement, the household borrows b in the credit market and arranges the transfer τ to repay the bank in accordance with the plan agreed. Equilibrium. We consider competitive equilibria with free entry in the credit market, and SPE in bargaining. Generically, an equilibrium is a situation where households choices maximize the expected value of utility given prices and bank s (bargaining) responses; banks choices maximize the expected recovered value given prices and households responses; and prices make banks (expected) profits equal to zero given households and bank s decision rules. More specifically, it will be instructive to break it down into two blocks, a no-bargaining equilibrium and a bargaining equilibrium. A no-bargaining equilibrium takes as given the bargaining values for the households and the banks, and determines the price schedule, and the values and decision rules for repayment, default and Chapter-7 bankruptcy. This is akin to the standard equilibrium in recent consumer bankruptcy models (that is, if with arbitrarily bad outcomes of bargaining). A bargaining equilibrium takes as given the values of being clean and under Chapter 7 and prices, and determines the bargaining policy rules and reservation values. An equilibrium is a combination of no-bargaining and bargaining equilibria that are consistent with each other. To make this definition operative, we move on to state formally the decision problems and conditions implied in an equilibrium of the model. 3 Equilibrium conditions and characterization To help notation, denote o the party that gets to propose a bargaining plan, and use ho to index situations when the household offers and lo for situations when the lender offers. Also, use ac to index situations where a offer is accepted and noac for when it is not. This can be expressed recursively.

8 bargaining in the shadow of chapter Households Households take as given the market price schedule for debt q and the bank s reservation value in the failed bargaining states Wo D,B conditional on the proposer s type o {lo, ho}. 4 We first consider in the next subsection the key decisions about whether and how to default under each of the states, for given values associated with these choices. Then in the following subsection we will characterize these underlying values of the default options. 3.. Default choices More specifically, the default decisions yield a number of functions of the state (a, b, s) for the household. Starting from a clean record z =, we have the probability of choosing Chapter 7 default conditional on the proposer p 7,o ; the values to defaulting conditional on the proposer Vo D ; the ex-ante value of defaulting V D and the corresponding default rate d; the value of a clean record V. On the other hand, starting from a situation of previous failed bargaining z = 2, 3, we have the probability of choosing Chapter 7 default conditional on the proposer p B D,B 7,o ; the values conditional on the proposer Vo. Consider first the households with a good history z =. Having chosen to default, they decide whether to choose Chapter 7 or bargaining. The probability of Chapter 7 and values are, for o = lo, ho, ( (V 7 (a, s) χ 7 ) (Vo B (a, b, s) χ B ) ) p 7,o (a, b, s) = CDF () V D o σ 7vsB (a, b, s) = ( p 7,o (a, b, s))(v B o (a, b, s) χ B ) + p 7,o (a, b, s)(v 7 (a, s) χ 7 ) (2) where V 7 and Vo B denote the bankruptcy and bargaining option values, respectively, and CDF denotes the cdf of a standard Normal. Note that the Vo B s are ex-post values known after uncertainty about the proposer is revealed. The ex-ante value is V D (a, b, s) = γ ho Vho D (a, b, s) + ( γ ho)vlo D (a, b, s) (3) Then the choice whether to default of repay yields the default probability ( V D (a, b, s) V R ) (a, b, s) d(a, b, s) = CDF σ DvsR (4) and ex-ante value V (a, b, s) = ( d(a, b, s))v R (a, b, s) + d(a, b, s)v D (a, b, s) (5) where V R stands for the value of repaying. 4 The superindex D, B indicates a state of default following a period of failed bargaining, that is one of z = 2, 3. With the subindex ho it corresponds to z = 2; with the subindex lo it corresponds to z = 3. Admittedly, this notation may not be the most fortunate one.

9 bargaining in the shadow of chapter 7 8 There is no decision about status for the household while bearing the bankruptcy flag z =. In the failed-bargaining states z = 2, 3, the household must decide whether to continue bargaining or file for Chapter 7. 5 The probability of bankruptcy and the resulting value are, for o = lo, ho, ( (V p B 7 (a, s) χ 7 ) V B ) lo (a, b, s) 7,o(a, b, s) = CDF (6) V D,B o 3..2 The option values σ 7vsB (a, b, s) = ( p B 7,o(a, b, s))v B o (a, b, s) + p B 7,o(a, b, s)(v 7 (a, s) χ 7 ) (7) The objects that all the above default choices take as given are the value of repaying V R ; the value of filing for bankruptcy V 7 ; the value of failed bargaining Vo B. The value of repaying in the clean state: V R (a, b, s) = max u (c) + βe [ V ( a, b, s ) s ] (8) b,a subject to c q ( a, b, s ) b + q a e (s) b + a φ(a /a) The value for the household to filing for Chapter 7 when z = or being in the bankrupt state z = is determined as follows. (Note that when z = the agent holds no debt so it must be that b =.) The household s value: V 7 { [ (a, b, s) = max a u(c) + βe π 7 V 7 (a,, s ) + ( π 7 )V (a,, s ) ]} s.t. c + q a = e(s)( θ 7 ) + a max(, min(a ā 7, b)) φ 7 (a (9) /a) Consider now the values of bargaining. Bargaining is over proposed transfers of current resources τ and debt swaps b. It also involves a choice for next period a. Suppose the proposer is the household. The value if the offer is not accepted (i.e., of delaying agreement by making an unacceptable offer) is V B ho,noac (a, b, s) = max{u(c) + δe[γ ho,ho V D,B ho (a, b, s ) + ( γ ho,ho )V D,B lo (a, b, s )]}, () a subject to the budget c + q a = e(s)( θ B ) + a φ B (a /a) min(τ b b, e(s)( θ B ) + a) This gives decision rule for a ho,noac (a, b, s). The value if the offer is accepted is Vho,ac B (a, b, s) = max [u(c) + δev a,b (a, b, s )], () subject to the budget c + q a = e(s)( θ B ) T (a, b, s, a, b ) + a φ B (a /a) where the transfer T (.) is such that T (a, b, s, a, b ) + b q(a, b, s) = min{q E[γ ho,ho W D,B ho (a ho,noac (a, b, s), b, s ) + ( γ ho,ho )W D,B lo (a ho,noac (a, b, s), b, s ) s], b} (2) 5 There is no extra non-pecuniary cost to continuing bargaining but there is still the cost to fling bankruptcy. We may or may not want to change this assumption.

10 bargaining in the shadow of chapter 7 9 Expression (2) says that the household must get the bank to accept by offering either fullrepayment or the bank s option value of postponing agreement. This yields the settlement plan a ho,ac (a, b, s), b ho,ac (a, b, s) and τ ho,ac(a, b, s) = T (a, b, s, a ho,ac (a, b, s), b ho,ac (a, b, s)). The household s value of bargaining is thus Vho B (a, b, s) = max{v ho,noac B (a, b, s), V ho,ac B (a, b, s)} (3) Suppose now the proposer is the lender. If the offered plan is not accepted V B lo,noac (a, b, s) = max{u(c) + δe[γ lo,lo V D,B a lo (a, b, s ) + ( γ lo,lo )V D,B ho (a, b, s )]} (4) with c + q a = e(s)( θ B ) + a φ B (a /a) min(τ b b, e(s)( θ B ) + a), which yields the policy rule a lo,noac (a, b, s). Clearly, the household s acceptance condition must hold as an equality at the plan proposed by the bank. Therefore, the household gets its reservation value Vlo,ac B (a, b, s) = V B lo,noac (a, b, s). The household s value of bargaining is thus 3.2 Banks Vlo B B (a, b, s) = Vlo,noac (a, b, s). (5) Banks make decisions when in a failed-bargaining default state z = 2, 3. In such a situation, the bank takes as given the market prices q, the household s option values of remaining in that state Vo,noac B for o = lo or clearing its record V, and the household s decision rules about bankruptcy p B 7,o and no-acceptance savings a o,noac for o = lo, ho. We first determine the value to the bank in such a state as a function of the bank s option values of bargaining Wo B for o = lo, ho and of bankruptcy W 7. This value is, for l = lo, ho, W D,B o (a, b, s) = ( p B 7,o(a, b, s))w B o (a, b, s) + p B 7,o(a, b, s)w 7 (a, s) (6) Second we determine the underlying option values. The recovered value in bankruptcy is { W 7 a ā 7 (a, b, s) = γ 7 min(a ā 7, b) a > ā 7 (7) Consider now the bargaining option when the household proposes. The bank s value to accepting is Who,ac B (a, b, s) = τ ho,ac(a, b, s) + b q(a ho,ac (a, b, s), b ho,ac (a, b, s), s), (8) and the value to rejecting is W B ho,noac (a, b, s) = q E[γ ho,ho W D,B ho (a ho,noac (a, b, s), b, s ) ( γ ho,ho )W D,B lo (a ho,noac (a, b, s), b, s ) s] + min(τ b b, e(s)( θ B ) + a) (9)

11 bargaining in the shadow of chapter 7 The bank would get its reservation value unless it exceeds full repayment so that Who,ac B (a, b, s) = (a, b, s), b}. Therefore the value of bargaining to the bank is then: min{w B ho,noac W B ho (a, b, s) = min{w B ho,noac (a, b, s), b} λb b (2) Consider finally bargaining when the bank proposes. The values to the bank to having the household accepting is Wlo,ac B (a, b, s) = max T (a, b, s, a, b ) + b q(a, b, s), (2) a,b subject to full repayment being the best the bank can propose with the transfer satisfying Wlo,ac B (a, b, s) b (22) T (a, b, s, a, b ) = q a + e(s)( θ B ) + a φ B (a /a) ( ( σ) [ V B lo,noac (a, b, s) βev (a, b, s ) ]) /( σ) (23) This last condition (23) says that the bank will make an offer that leaves the household indifferent between accepting and delaying the outcome. This yields the settlement decision rules a lo,ac (a, b, s), b lo,ac (a, b, s) and τ lo,ac(a, b, s). The value if the offer is not accepted: W B lo,noac (a, b, s) = q E[γ lo,lo W D,B lo (a lo,noac (a, b, s), b, s )+ ( γ lo,lo )W D,B ho (a lo,noac (a, b, s), b, s ) s] + min(τ b b, e(s)( θ B ) + a) (24) The bank s value of bargaining here is therefore 3.3 Zero profits Wlo B B B (a, b, s) = max{wlo,noac (a, b, s), Wlo,ac (a, b, s)} λb b (25) With fee-entry competition in the credit market, the prices of debt imply zero ex-ante profits for banks extending loans. The evaluation of cash flows in the coming periods depends on the households default rule d and the banks ex-ante value in the event of default W D. Given these, the zero-profit condition for the lender yields the price schedule q ( a, b, s ) = ( π ( a, b, s ) λ ) q + q s Γ s,s d(a, b, s )W D (a, b, s )/b (26) where the probability of default is π(a, b, s) = s Γ s,s d(a, b, s ).

12 bargaining in the shadow of chapter 7 It remain to determine the ex-ante value of the bank under default with a clear record D D. It depends on the clean households decision rules about bankruptcy p 7,o and the bank s option values of bargaining Wo B, for o = lo, ho, and of bankruptcy W 7. The ex-ante recovered value when there is default conditional on the proposer o = lo, ho is W D o (a, b, s) = p 7,o (a, b, s)w 7 (a, b, s) + ( p 7,o (a, b, s))w B o (a, b, s) (27) The ex-ante value is therefore 3.4 Formal definition W D (a, b, s) = γ ho Who D (a, b, s) + ( γ ho)wlo D (a, b, s) (28) Define first an equilibrium for given bargaining outcomes. Given the bargaining values for the household Vho B and V lo B, and for the bank W ho B and W lo B, a bargaining-conditional equilibrium is a a price schedule, value functions and policy rules such that: (i) Given the value of bankruptcy for the household V 7, the value of defaulting V D and the probabilities of bankruptcy, p 7,lo and p 7,ho, solve () to (3). (ii) Given the ex-ante value V and the price schedule q, the value of repaying V R solves (8). (iii) Given the ex-ante value V, the value of bankruptcy V 7 solves (9). (iv) Given the values to repaying V R and defaulting V D, the ex-ante value V and default probabilities d solve (5) and (4). (v) The bank s value under bankruptcy W 7 solves (7). (vi) Given the probabilities of bankruptcy, p 7,lo and p 7,ho, and W 7, the bank s value to defaulting W D lo, W D ho and W D solve (27) and (28). (vii) Given default probabilities d and bank s default value W D, the price schedule q solves (26). Given q, (i)-(vi) defines a fixed pint problem in V, V 7, V D, V R, W 7 and W D and the associated decision rules. Adding (vii) results in another fixed point problem in q. These are for given bargaining values V B o and W B o. A bargaining equilibrium. Given non-bargaining functions V, V 7, W 7, and prices q a bargaining equilibrium are values and decisions functions such that:

13 bargaining in the shadow of chapter 7 2 (i) Given the values of continuing bargaining for the household Vlo B and V ho B, the probabilities of filing for bankruptcy or bargain and the values, p B 7,lo, pb 7,ho, V D,B lo and V D,B ho, solve solve (6), (7). (ii) Given the the values to continue bargaining for the household, V D,B lo and V D,B ho, and for the bank, W D,B lo and W D,B ho, the bargaining value for the household when it proposes,, solves () to (3). V B ho (iii) Given the the values to continue bargaining for the household, V D,B lo and V D,B ho, the bargaining value for the household when the bank proposes, Vlo B and V lo,noac B, solves (4) to (5). (iv) Given the probabilities of filing for bankruptcy or bargain and bank values, p B 7,lo, pb 7,ho, and Wlo B, W ho B D,B, the values for the bank of being in default and bargaining Wlo and W D,B ho solve (6). (v) Given the values for the bank of being in default and bargaining W D,B lo and W D,B ho and the household s behaviour if its proposal is not accepted Vho,noac B, the banks value of entering bargaining when the household proposes Who B solves (9) and (2). (vi) Given the values for the bank of being in default and bargaining W D,B lo and W D,B ho and the household s behaviour under Vlo,noac B, the banks value of entering bargaining when the banks proposes Wlo B solves (2), (22), (23), (24), and (25). A bargaining equilibrium if a fixed point in Vo B values V, V 7, and W 7, and q. and W B o. It takes as given non-bargaining An equilibrium is functions Vo B, Wo B V, V 7, W 7 and q that satisfy both a bargaining equilibrium and a no-bargaining equilibrium. Finding an equilibrium involves solving a number of nested fixed point problems. One possible lay out, used in the computation, is as follows.... We can show two theoretical results. First taken, debt prices as exogenous we can show that a unique bargaining equilibrium exists. Second under limited conditions we can show that an equilibrium to the entire model exists. 4 Calibration Strategy. The model has a large number of parameters. In order to facilitate the task of assigning values, we will first fix most parameters so that they are broadly consistent with observations on debt (and asset) holdings. Then the remaining few parameters will be chosen in order to match a limited number of more specific targets related to default an interest rates. Various of the parameters fixed initially will later have to be considered for

14 bargaining in the shadow of chapter 7 3 sensitivity analysis. Fixed parameters. A model s period corresponds to one year. Consider the parameters set directly. The discount rate β and risk aversion σ are values in line with usual choices in the literature (but see more on β later). The price of q is set to accord with a 3 per cent risk free interest rate. The intermediation costs λ matches existing estimates of 4 per cent. We assume away asset losses under Chapter 7 which dictates the choice for γ 7. The exemption level for under bankruptcy varies widely across U.S. states. The choice for ā 7 is equivalent to exempting below average income. The income process is described by two possible states, with relative realizations of the endowment of 9/, and (to simplify) identical and independent transition probabilities. We will also set the adjustment cost parameter φ. For lack of better information, the parameters related to the identity of the proposer γ ho, γ ho,ho and γ lo,lo will also be fixed. Nonetheless different economies will be considered that make different assumptions in this regard. For the time being we only consider the case where the proposer is always the lender. Table. Parameters set directly Description Parameter value Risk aversion σ 2. Discount β.88 Bank s cost λ.4 Capital loss Ch. 7 γ 7. Asset exemption ā 7.5 Risk-free interest /q.3 Number inc. types N e 2 Value of inc. e..9 Transition types Γ e,e Adjust. cost φ.5 Proposer γ ho. γ ho,ho, γ lo,lo.. Endogenous targets. The interest of this exercise is in measures of indebtedness, terms of credit and default. The remaining endogenous parameters will be aiming to produce outcomes consistent with the following observations. The unsecured debt to income ratio for households with debt falls between 4 and 5 per cent (SCF 27, Tables and 3) and the to income ratio is about.5. As for debt defaults, one could tentatively take the number of households not current on their debt repayments by over two months. In the Survey of Consumer Finances the figure stands at about 3.5 per cent. Regarding the portion of defaults that take place under the formal bankruptcy procedures (i.e., Chapter 7), this is close to per cent of the population. The average interest rate on credit card balances is between 2 and 3 per cent.

15 bargaining in the shadow of chapter 7 4 Endogenous parameters. For given values of all the previous parameters, the choice of the collection tax τ b, the time cost of Chapter 7 bankruptcy π 7 and the utility costs to defaulting χ d seeks to match the targets for the overall default rate, the mix between Chapter 7 and bargaining, and the average lending interest rate. Note also that the variance of the noises, σ 7vsB and σ DvsR, will need to be set to perform their designed technical function. We start with a zero collection tax. (We will relax this later.) Table 2. Endogenous parameters Description Parameter value Observable Prob. bankrupt π 7.68 def. rate.368 Collection tax τ b. def. Ch7.6 Default costs χ d.6 interest /.843-=.86 Noises σ 7vsB.35 debt/inc.62 σ DvsR.2 The choice of parameters yields implications fairly close to the empirical targets. The interest rate might be too high thus reflecting the tension with having to meet the overall default rate and given the assumption of zero collection tax. Regarding other variables, the proportion holding positive unsecured debt is.335 to be compared with.46 in SCF Findings This section describes the benchmark economy, with a focus on the patterns for formal and informal default across different households. It will also discuss the consequences of changes in some key parameters. 5. Default outcomes and credit terms The next Table 3 displays some summary figures regarding the mass of households in different states and decisions. The households who default to engage in bargaining do predominantly accept the lender s first proposed plan and settle immediately. About 7 per cent of those who settle will repeat defaulting next period. There is only a small proportion of bargaining defaulters who turn down that offer and do not settle, starting next period with bankruptcy state z = 3. The majority of households about 9 per cent in such state will carry on bargaining, settling or rejecting in about the same measure; the remaining households file for formal Chapter 7 bankruptcy. Since the mass of households in a bargaining state z is negligible, in what follows we will focus on variables related to clean households with z =.

16 bargaining in the shadow of chapter 7 5 Table 3. Mass of households by state and choice Description Mass Clean households z = : Number of z =.9753 With positive debts With positive debts and Defaulters Defaulters to ch Defaulters to B settle Defaulters to B reject.232 Repeat defaulters.7366 Bargaining households z = 3: Number of z = Number to Ch7.322 Number to B & settle.9 Number to B & reject.999 The top part of Table 4 reports averages of debt and asset holdings for clean households making different choices. Households who default hold more debt and fewer than the average household (whether or not conditioning for having positive debt). Among the defaulters, those who formally file under Chapter 7 hold smaller positions of both debts and, and have a lower net worth. The flip side is that bargainers hold more and net worth. Table 4. Portfolios and terms Holdings: Description debts All With positive b > Defaulters to Ch Defaulters to B Decisions: Description borrow save price transfer recovered Non defaulters Defaulters to Ch Defaulters to B How do portfolios change depending of the default choice? Table 4 also shows the borrowing and saving average decisions and terms of credit for the different groups of households. Households filing under Chapter 7 experience a sight reduction in. Bargainers experience a sharp reduction in their gross financial positions. The settlements reached involve reductions in debt and of 9 and 4 per cent respectively, implying a recovery rate of about 2 per cent of debt outstanding.

17 bargaining in the shadow of chapter 7 6 In sum, bargaining appears to be associated with default by high net worth households, with settlements involving substantial reductions in debt and. In order to analyze this broad brush picture, we turn now to showing where in the distribution of households the different choices are made and the policy rules of the households involved. 5.2 The distribution of default Figure displays the distribution of households over and debt. Nearly all households with positive debts also hold a positive level of. Agents with low levels of debt, from. to.3, are spread over a wide range of asset positions. At moderate levels of debt, such as.4, asset holdings are concentrated in the lower range. Above that debt level, there is a long gap with zero mass until a substantial number of households emerges with high debt levels of.2 and.3 and intermediate level of of.6 and.65., respectively mass.2.2 mass s= debt debt.4 Figure : Wealth distribution. Low/high income in left/right panel. Where does default happen? The top section of Figure 2 represents the policy rule for default, d, over asset and debt levels. The low income households do default on a wider range of positions. There are areas with positive mass of households in debt from Figure where the rule implies positive default. Specifically, the bottom section of Figure 2 shows the distribution of the mass of households who default. Two differentiated types of households fail to repay, one with intermediate debt (between.2 and.4) and low (between and.2), and another type with high debts (between.2 and.3) and high (between.6 and.65). Of these households, all of those with a low income realization default per cent. Therefore, that most default happens in the high- group only reflects the size distribution of these types from Fig. In contrast, high income households only default partially, with the low-asset group accounting for most of it. Nonetheless, the scale of default for the low income group is orders of magnitude bigger so, overall, most 6 Because the income process is iid and symmetric, the distribution is the same for the two income groups. This helps exposition.

18 bargaining in the shadow of chapter def. s=.5 def. s= debt debt mass defaulters s=.8.6 mass defaulters s= debt debt.4 Figure 2: Default. Policy function (top) and mass distribution (bottom). Low/high income in left/right panel. default occurs in the high-asset type of defaulting households. Turning to the default mix between formal Chapter 7 and informal bargaining, the top section of Figure 3 shows the decision rule for the fraction of defaulting households that choose Chapter 7 instead of bargaining. The preference for Chapter 7 is U-shaped in the level of, with a sharp drop near the asset exemption level, especially at high levels of debt. The preference for Chapter 7 is mainly decreasing in the level of debt, except possibly at low levels of. These patterns for the preference over the default mix combined with the characterization of the distribution of overall default result in a picture for the distribution of the default mix. The low-asset defaulters choose predominantly Chapter 7, whereas the high-asset defaulters opt in favour of bargaining. Specifically for low endowment households, between 75 and 8 per cent of low-asset defaulters choose Chapter 7; between 98 and 99 per cent of high-asset defaulters choose bargaining. The mass distribution of bargainers is explicitly shown in the bottom panel of Figure 3, to be compared with the analogous representation for all defaulters earlier in Figure 2. Bargainers make up for the immense majority of the defaulters of the high-asset type, but they are only a small proportion of

19 bargaining in the shadow of chapter debt % of default that is Ch7 % of default that is chapter debt bargainers s=.8.6 bargainers s= debt debt.4 Figure 3: Default mix. Policy function for probability of Chapter 7 (top) and mass distribution of bargainers (bottom). Low/high income in left/right panel. the defaulters of the low-asset type. 5.3 Bargaining or Chapter 7 The households who default via bargaining have a distinctive type of portfolio, holding debt markedly above the rest of population and with on a narrow range above the bankruptcy exemption level. Why do these households prefer bargaining? With that can be seized under Chapter 7 to pay off a considerable outstanding debt, the option to bargain is more attractive. There the household can issue debt and hold on to its to smooth consumption instead of having to disinvest hurriedly. Figure 4 shows, for situations where bargaining is the optimal choice, the difference between the savings decision rule in the case that households bargain and settle and in the case where households file under Chapter 7. By choosing to bargain, the household keeps more than under Chapter 7 bankruptcy.

20 bargaining in the shadow of chapter savings B vs Ch debt Figure 4: Savings when bargaining minus savings if Chapter 7. Low income only. Why is there any Chapter 7 at all? Chapter 7 occurs in the situations shown in Fig. 2 and 3 where households have little in so the discharge of debt precludes the bank from grabbing anything from the household. Bargaining, however, means the bank can always seize something until a settlement is reached and becomes a suboptimal choice for the household. The top section of Fig. 3 also shows that households may prefer Chapter 7 at high enough levels of, although this does not materialize in the stationary distribution. When are sufficiently large, changes in its level have no effect on the cost of formal bankruptcy since debt is repaid in full under Chapter 7. However, the net cost of bargaining is increasing in the household s. The household is more willing to sacrifice current consumption in order to regain access to credit should income fall in the future. This weakens the household s bargaining reservation value. In effect, it can be seen (graph not shown) that recovery rates rise noticeably with the household s level of. This makes the relative preference for Chapter 7 stronger for sufficiently high levels of. How do bargaining households end up in the characteristic position of high debt and? We need to examine the portfolio decision rules for clean households. The policy function for savings, shown in the top section of Figure 5, has the usual properties, not being very sensitive to debt level. The policy function for borrowing is in the bottom section of Figure 5. Note that for the high income realization s = 2 borrowing is pretty thin. The bargainers state is necessarily reached after one period of low income s = so we focus on this situation. For portfolio states where new savings is below the exemption level the

21 bargaining in the shadow of chapter saving a' for s= savings a' for s= debt debt b' for s=..8.6 d borrowing b' for s= debt debt.4 Figure 5: Decision rules for savings (top) and borrowing (bottom) for non defaulters. Low/high income in left/right panel. household borrows only small amounts. For some higher level of wealth leading to savings narrowly above the exemption level, the household comes to borrow heavily. At even higher wealth, borrowing becomes small again. The states with savings above the exemption level and associated with a surge in borrowing are those leading to default with bargaining. Quite a lot of debt in the economy is accounted for by this group of households. In sum, bargainers are low-income agents who in the previous period made some savings above the bankruptcy exemption level. Specifically, this happens from states in the distribution with the /debt pairs (.85,.), (.85,.), (.9,.). Why the distinctively large debts the bargaining households carry? These households anticipate they will default and bargain if a low income realization strike again. The level of debt will thus not affect negatively the outcome of the negotiations, so this household issues whatever debt is needed to maximize the amount of current consumption from borrowing. This is also true of low- Chapter 7 defaulters. The difference is that under bargaining banks anticipate they will recover something so this debt commands a less steep price schedule compared with that faced by households bound to declare Chapter 7 bankruptcy.

22 bargaining in the shadow of chapter price of debt (s=) debt Figure 6: Debt prices. Low income (left) and high income (right). These consumers can therefore issue a larger level of debt in order to increase consumption. Figure 6 displays the schedule of debt prices for low income households to demonstrate the point. 7 The schedule is much flatter at high debt positions associated with bargaining default than al low debt positions associated with Chapter 7 default. 5.4 Delay There are situations of bargaining where a settlement is not achieved immediately. Figure 7 shows the distribution of the outcome whether the plan proposed is accepted. For low income households, the decision is always to accept (value ) in the regions where there is positive mass of debt. For high income households, the outcome is not to accept (value ) in the region of high debts where with positive mass of debt and bargaining. The lender proposes a plan that is not acceptable because it prefers to delay proposing an acceptable plan. The reason is that the high earner will be saving while a settlement is not reached and the lender will be able to increase the recovery rate when the household has more. Figure 8 shows that the recovery rate in a settlement will increases sharply with the households asset position in the debt relevant region, especially if the household continues to have a high income realization. 7 High income households face exactly the same set of prices because of the assumed income process.